Capitalism works by contract. People only make agreements if each expects to benefit. Therefore under capitalism, anyone who wants to accumulate as much wealth as possible must do so by finding the best ways to increase the wealth of other people as well (i.e. providing what consumers demand). Capitalism forces wealth to increase in all parts of society, not just one.

But there's still the possibility of one person gaining faster than others, so let's consider a situation where there's huge disparity, and more and more of the economy falls under the control of one big firm. There's an important part of Austrian Economic theory known as the "economic calculation problem", which as it was originally formed was used to critique socialism, where the government would plan the entire economy, but it works equally well for a situation where one firm would try to own everything.

The argument goes that to act economically you need to be able to tell if a project is profitable, that is to say making sure you aren't wasting resources on something people value less than some other thing. The problem though is that we have a bunch of different kinds of inputs that can produce a bunch of different kinds of outputs and lots of different combinations of inputs that can make the same output. Since they are all different though, how can we compare them? We can't add and subtract apples from oranges! The market answer for this is using money prices. We can estimate our costs and our benefits in the terms of a single good, money, and then add and subtract them from each other to tell whether we made a profit or a loss. This allows people to estimate what projects are profitable or not beforehand and acting accordingly.

The reason socialism sucks then is because if the government owned all inputs, then it follows that these inputs are not traded in a market. If there's no market for these goods, there are no prices. If there are no prices, then you can't sum up your costs in terms of money. It follows then that socialism is doomed to fail because it can't use economic calculation to figure out what's a good idea or not. The same follows for "one big firm" if wealth began to accumulate in the hands of a few people who owned everything, these people would quickly begin to fail in the market place and start to lose money, not being able to tell what to do or what's profitable.

Good question, thanks for asking!